Abstract: By the Uniformization Theorem a compact Riemann surface other than the Riemann Sphere
or an elliptic curve is uniformized by the unit disk and equivalently by the upper half plane. The upper
half plane is also the universal covering space of the moduli space of elliptic curves equipped with an
appropriate level structure. In Several Complex Variables, the Siegel upper half plane is an analogue
of the upper half plane, and it is the universal covering space of moduli spaces of polarized Abelian
varieties with appropriate level structures. The Siegel upper half plane belongs, up to biholomorphic
equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research
is taking place. Especially, finite-volume quotients of bounded symmetric domains, which are naturally
quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic
Geometry, Arithmetic Geomtry and Number Theory, and an important topic is the study of uniformizations
of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved from methods
of Diophantine Geometry, Model Theory, Hodge Theory and Algebraic Geometry for Shimura varieties,
techniques for the general case of not necessarily arithmetic quotients have just begun to be developed.
We will explain a differential-geometric approach to the study of such algebraic subsets revolving around
the notion of asymptotic curvature behavior and the use of rescaling arguments, and illustrate how this
approach using transcendental techniques leads to various characterization results for totally geodesic
subvarieties of finite-volume quotients without the assumption of arithmeticity. Especially, we will explain
how the study of holomorphic isometric embeddings of the Poincare disk and more generally complex unit
balls into bounded symmetric domains can be further developed to derive uniformization theorems for
bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the
universal covering map.
